The "Traveling Salesman Problem" (TSP) is a classic optimization problem in the field of operations research and computer science. At its core, the TSP poses the following question: Given a list of cities and the distances between each pair of cities, what is the shortest possible route that a traveling salesman can take to visit each city exactly once and return to the original city?

To further explain:

Complexity: While the problem sounds simple, it becomes increasingly complex as more cities are added. For instance, with 4 cities, there are only 24 possible routes (4 factorial, or 4!), but with 10 cities, there are 3,628,800 possible routes (10!). As the number of cities (or nodes) grows, the number of possible routes grows factorially, making it computationally challenging to find the optimal solution using brute force (i.e., checking every possible route).

Applications: Though it's often framed in terms of a salesman and cities, the TSP has applications in various domains, including logistics, vehicle routing, and circuit design. Any problem that requires optimizing a sequence of stops can be framed as a variant of the TSP.

Solution Approaches: There are various algorithms and heuristics that have been developed to tackle the TSP. Exact algorithms guarantee an optimal solution but can be time-consuming for large numbers of cities. On the other hand, heuristic and metaheuristic approaches (like genetic algorithms, simulated annealing, and nearest neighbor) provide good solutions in shorter times but don't always guarantee the absolute best route. Solvice uses a fusion of Heuristics, Optimization and Machine Learning to solve the Travelling Salesman Problem.

In essence, the TSP is a foundational problem in optimization that encapsulates the challenges of making efficient decisions in the face of combinatorial explosion and has spurred a wealth of research and methodologies to address it.